THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF
MONOMIAL CURVES
K. RANESTAD∗
Abstract. A monomial curve is a curve parametrized by monomials. The degree of
the secant variety of a monomial curve is given in terms of the sequence of exponents
of the monomials defining the curve. Likewise, the degree of the join of two monomial
curves is given in terms of the two sequences of exponents.
Contents
1. Introduction
2. The multiplicity sequence of a plane curve singularity
3. The degree of the secant variety of a monomial curve
4. The degree of the join of two monomial curves
References
1
2
5
7
14
1. Introduction
A monomial curve C is the image of an injective morphism of f : P1 → Pr defined by
monomials. After ordering the monomials by ascending degree it is therefore given by
[s : t] 7→ [sd : sd−a1 ta1 : . . . : sd−ar−1 tar−1 : td ]
where a1 < a2 < . . . < ar = d. So this latter sequence completely determines C. We define
the first secant variety SecC to be the closure of the union of lines that meet C in two
distinct points. The first aim of this note is to compute the degree of this secant variety
as a subvariety of Pr . According to a well-known argument using a general projection
π : C → C ⊂ P2 this degree is given by the formula
d−1
degSecC =
− δp − δq
2
where δp and δq are the genus contributions of the cusps at p = π([1 : 0 . . . : 0]) and
q = π([0 : . . . : 0 : 1]) on C. Equivalently, 2δp and 2δq are the Milnor numbers of the
cusps at p and q. To compute δp and δq given C, we analyze the Puiseux expansions of
C at the cusps, and apply an algorithm due to Chisini and Enriques, eventually refined
and given a closed form by Casas-Alvero.
Date: September 14, 2005.
1991 Mathematics Subject Classification. Primary 14; Secondary 14.
∗
Supported by MSRI.
1
2
K. RANESTAD
r
Given two curves C and D in P we define their join Join(C, D) to be the closure of
the union of lines that meet C and D in two distinct points. We consider the join of two
monomial curves C and D: In the notation of the previous section we ask that the two
curves are defined by
C : [s : t] 7→ [sdC : sdC −a1 ta1 : . . . : sdC −ar−1 tar−1 : tdC ]
where a1 < a2 < . . . < ar = dC , and
D : [s : t] 7→ [sdD : sdD −b1 tb1 : . . . : sdD −br−1 tbr−1 : tdD ]
where b1 < b2 < . . . < br = dD . Again the two sequences
a1 < a2 < . . . < ar = dC ,
b1 < b2 < . . . < br = dD
determine the two curves completely, and our second goal is to compute the degree of
the join of C and D as a subvariety of Pr . In this case the general projection of the two
curves to P2 gives the formula
degJoin(C, D) = dC · dD − Is (C, D),
where Is (C, D) is the sum of the intersection multiplicities in P2 of the two curves at the
images of intersection points between the two curves in Pr . Algorithms computing the
sum of intersection multiplicities Is (C, D) are given in section 4.
The author thanks MSRI for excellent working conditions, Bernd Sturmfels for his
inspired interest in the problem, Eduardo Casas-Alvero for giving the solution a nicer
form and the referee for pointing out inaccuracies in an early version.
2. The multiplicity sequence of a plane curve singularity
A crucial ingredient in the two algorithms below is the multiplicity sequence of a plane
curve singularity. Given a point p in the plane and a sequence of blowups at simple
points (p = p0 , p1 , p2 , ..., ps ), such that all exceptional divisors lie over p, i.e. is mapped
to p by the natural map to the original plane, and such that the strict transform of the
curve is smooth. The multiplicities m0 (C), (resp. mi (C), i > 0) of C at p (respectively
its strict transforms at pi ), form the multiplicity sequence of C at p with respect to the
sequence of blowups. Equivalently, the multiplicity sequence coincides with the sequence
of intersection numbers of the strict transform of C with the exceptional divisor of each
blow up. The multiplicity sequence may contain 1’s, but these would not appear in a
blowup that provides a minimal resolution of the singularity. In the latter case we say
that the multiplicity sequence is minimal. Note that by the unicity of a minimal resolution
of a plane curve singularity, the minimal multiplicity sequence is unique. Both minimal
and nonminimal cases will however occur in our setting.
We shall use the multiplicity sequence of plane singularities with given Puiseux series.
Consider the parameterized affine plane curve
C : t 7→ (tm , tk1 + tk2 + · · · ).
This plane curve has a cusp at the origin, where t = 0. The multiplicity sequence is
computed from the sequence m, k1 , k2 , . . . as described in a result of Enriques and Chisini
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
3
[1] Theorem 8.4.12. This algorithm is presented below, and the genus contribution δ at
t = 0 is subsequently computed from the multiplicity sequence.
Algorithm 2.1. (Multiplicity sequence.)
Consider the strictly increasing sequence
m < k1 < k2 < . . .
Step 1. The gcd-sequence and characteristic terms. (This step is not necessary to
compute the multiplicity sequence, but clarifies the role of the different terms ki .) Let g0 =
m and gi = gcd{m, k1 , . . . , ki } for i > 0. The gi form the gcd-sequence of m, k1 , . . . , kr :
g0 ≥ g1 ≥ g2 ≥ g3 . . . .
Clearly, in the gcd-sequence, gi = 1 for some i, since otherwise the parameterization is
not 1 : 1. The characteristic terms in the sequence k1 , k2 , . . . are the terms
ki1 , ..., kis
where i1 = min{i|gi < m}, i2 = min{i|gi < gi1 } etc. Thus m = g0 > gi1 and
gi1 > ... > gis = 1.
In particular the number of characteristic terms is finite and bounded by the number of
prime factors in m.
Step 2. Given the sequence
m < k1 < k2 < . . .
let κi = ki − ki−1 where k0 = 0 and i = 1, 2, .... We call
κ1 , κ2 , . . .
the difference sequence of the cusp. In our applications we will always have a finite
number of terms in this sequence, so we assume we have a difference sequence with s
terms.
Apply the Euclidean algorithm successively to the elements of the difference sequence:
Let
κi = ei,1 ri,1 + ri,2
ri,1 = ei,2 ri,2 + ri,3
...
ri,w(i)−1 = ei,w(i) ri,w(i)
with 0 ≤ ri,j+1 < ri,j and r1,1 = m, ri,1 = ri−1,w(i−1) , i > 1. Note that
ri+1,1 = gi = gcd(m, k1 , ..., ki ).
The multiplicity sequence of the sequence {m < k1 < k2 < . . .} is ei,j times the
multiplicity ri,j , with 1 ≤ i ≤ s and 1 ≤ j ≤ w(i).
4
K. RANESTAD
We write the multiplicity sequence in the order it is computed, and with repetitions in
stead of the numbers ei,j . Note that the overall sequence is nonincreasing. The genus
contribution or δ-invariant of the sequence is given by
X ri,j δ=
ei,j
.
2
i,j
This sum is given a closed form in terms of the original sequence and its gcd-sequence
by the following result due to Casas-Alvero:
Proposition 2.2. Given the sequence
m < k1 < k2 < . . .
Let
g0 ≥ g1 ≥ g2 ≥ g3 . . .
be its gcd-sequence. Then the δ-invariant of the sequence is
1 X
δ= (
ki (gi−1 − gi ) − m + 1).
2 i≥1
In particular, the δ-invariant depend only on the characteristic terms of the sequence
m, k1 , k2 , . . ..
Proof. See [2, p. 194, ex. 5.6]. First, we may assume that the difference sequence is finite,
say with s terms. From the Euclidean algorithm applied to the elements of the difference
sequence we note that
w(i)
X
w(i)−1
ei,j ri,j = ki − ki−1 +
j=1
X
(ri,j − ri,j+1 )
j=1
= ki − ki−1 + ri,1 − ri,w(i) = ki − ki−1 + gi−1 − gi .
Secondly,
w(i)
X
w(i)−1
2
ei,j ri,j
= (ki − ki−1 )ri,1 +
X
j=1
(ri,j ri,j+1 − ri,j+1 ri,j )
j=1
= (ki − ki−1 )gi−1 .
Thus
w(i)
s X
X
ei,j ri,j = ks + g0 − gs ,
i=1 j=1
w(i)
s X
X
i=1 j=1
and
2δ =
s
X
i=1
2
ei,j ri,j
=
s
X
(ki − ki−1 )gi−1
i=1
(ki − ki−1 )gi−1 − ks − m + 1
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
=
s
X
5
ki (gi−1 − gi ) − m + 1
i=1
3. The degree of the secant variety of a monomial curve
Let C ⊂ Pr be a monomial curve defined by the sequence of positive integers a1 <
a2 < . . . < ar = d as above. Consider the secant variety SecC of C. This is a threefold,
so its degree is counted by the intersection of this variety with a general codimension
three subspace, or equivalently by the number of ordinary double points of the general
projection π : C → P2 . For a general projection the only other singularities on C = π(C)
are possible cusps at the image of the points π(p) and π(q) where p = [1 : . . . : 0] and
q = [0 : . . . : 1] in Pr . The formula for the arithmetic genus of a plane curve of degree d
and the computation of the genus contribution at these cups provides a formula for the
degree of SecC.
Proposition 3.1. Let C ⊂ Pr be a monomial curve defined by the sequence of positive
integers a1 < a2 < . . . < ar = d. Let bi = d − ar−i , for i = 1, ..., r − 1 and br = d. Let
gi = gcd(a1 , ..., ai ) and hi = gcd(b1 , ..., bi ), then
X
d−1
1 X
ai+1 (gi − gi+1 ) − a1 +
bi+1 (hi − hi+1 ) − b1 ) − 1.
degSecC =
− (
2 i
2
i
Proof. The arithmetic genus p(C) for a curve C on a smooth surface S is given by the
adjunction formula [3] on the surface:
2p(C) − 2 = C · C + C · KS
where KS is the canonical divisor on S. If C has multiplicity m at a point q on S, and
S 0 → S is the blowup of S at q, then the adjunction formula on S 0 says
2p(C 0 ) − 2 = C 0 · C 0 + C 0 · KS 0 =
= (C ∗ − mE) · (C ∗ − mE) + (C ∗ − mE) · (KS + E) = 2p(C) − 2 − m2 + m
where E is the exceptional divisor and C ∗ is the total transform and C 0 is the strict
transform of C (cf. [3] chapter V). Thus
m
0
p(C ) = p(C) −
,
2
so m2 is the genus contribution of a point of multiplicity m. After resolving all singularities on C ⊂ P2 by a series of blow ups centered at singular points of C or its strict
transform, the arithmetic genus of the strict transform C 0 is 0 since it is a rational curve.
At the ordinary double points the difference between
the arithmetic genus of the curve and
2
its strict transform after blowing up the point is 2 = 1. The points π(p) and π(q) are the
only other singularities on C. The contribution δp is by definition the difference between
the arithmetic genus of C and a strict transform that is smooth at the inverse image of
π(p) and isomorphic to C outside the point π(p). Likewise for δq . Since KP2 ∼
= −3L, where
6
K. RANESTAD
L is a line in the
plane, the arithmetic genus of C is given by 2p(C) − 2 = dC (dC − 3),
i.e. p(C) = d−1
. Adding all genus contributions we get the formula:
2
d−1
degSecC =
− δp − δq ,
2
P mi −1
P ni −1
where δp =
,
δ
=
and {m1 , m2 , . . .} and {n1 , n2 , . . .} are the multiplicity
q
2
2
sequences of C at π(p) and π(q) respectively.
The algorithm 2.1 computes these multiplicity sequences from the exponents of the
Puiseux expansion. By Proposition 2.2 it is enough for this algorithm to know the characteristic terms of the Puiseux expansion. Therefore we need only to find these terms
of the Puiseux expansion of C ⊂ P2 at π(p) and π(q). The projection π : C → P2
is determined by the choice of three projective coordinates (X : Y : Z) in Pr , two of
which, say X, Y vanish at p, and two, say X, Z, vanish at q. In particular in terms of the
parameterization of C ⊂ P2 we may choose
X = ta1 + b13 ta3 + ... + b1(r−1) tar−1 ,
Y = ta2 + b23 ta3 + ... + b2(r−2) tar−2 + tar ,
Z = 1 + b32 ta2 + b33 ta3 + ... + b3(r−2) tar−2 .
The coefficients bij are independant and determine the projection. Clearly the characteristic terms are determined by the projection, and therefore by the bij . The meaning of
”general” in general projection is that the characteristic terms are constant for an open
set of choices of coefficients bij . The following lemma says that a particularly simple choice
of coefficients belong to this open set, so that the characteristic terms can be computed
from this choice.
Lemma 3.2. The characteristic terms in the Puiseux expansion of C at p for a general
projection coincides with the characteristic terms in the Puiseux expansion
x = ta1 , y = ta2 + ta3 + ... + tar .
Proof. The characteristic terms of the latter Puiseux expansion is precisely the characteristic terms of the sequence a1 , a2 , . . . , ar as computed by the algorithm 2.1. So we compare
these characteristic terms with those in a Puiseux expansion of a curve parameterized by
X = ta1 + b13 ta3 + ... + b1(r−1) tar−1 ,
Y = ta2 + b23 ta3 + ... + b2(r−2) tar−2 + tar .
This Puiseux expansion is computed by a formal quotient
β1
β2
X
Y
and has the form
βn
t α + c2 t α + . . . cn t α . . . .
For our purposes it can also be done step by step, by iterated reparameterizations substituting t with t + utk for suitable u and k. We want to compare the characteristic terms
of the sequence α, β1 , β2 , ... with those of the sequence a1 , a2 , . . . , ar .
Since the characteristic terms are finite in number there is a largest one, say N0 . Clearly
then the curve parameterized by
X = tα + bN tN + . . . ,
Y = tβ1 + b2 tβ2 + ...,
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
7
with N ≥ N0 + α, has the same characteristic terms as C. So it is enough to find
a reparameterization of this kind. We do this step by step and reparameterize C by
substituting t with t + uta3 −a1 +1 for suitable u to cancel the coefficient of ta3 in X. In the
new parameterization we get:
0
0
X = ta1 + b014 ta4 + ... + b01r tar0 ,
Y = ta2 + b23 ta3 + ... + b2r tar + c1 tb1 + ....
where a04 > a3 , and all new exponents appearing are of the form ai + k(a3 − a1 ) for some
positive integer k. Compare the greatest common divisors gi , i = 1, 2, ... of a1 and the
i lowest exponents of t occurring in Y , before and after the reparameterization. The
difference is a possible repetition of some terms, and some possible cancellations. We
similarly reparameterize until the second exponent of t in the X-coordinate is bigger than
N0 + a1 and conclude that the characteristic terms of sequence
a1 , a2 , . . . , ar
include the characteristic terms of the general projection C. But specialization can only
result in fewer characteristic terms, so the inclusion must be an equality.
Since non-characteristic terms do not contribute to the δ-invariant the proposition
follows from Proposition 2.2.
Example 3.3. Consider the monomial curve C given by the sequence (0, 30, 45, 55, 78).
At p = [1 : 0], we may compute the δ-invariant from the Puiseux expansion with exponents
m = 30, (a3 , a4 , a5 ) = (45, 55, 78) The gcd-sequence is (30, 15, 5, 1) and the δ-invariant is
1
δp = (45(30 − 15) + 55(15 − 5) + 78(5 − 1) − 30 + 1) = 754.
2
At q = [0 : 1] we compute the δ-invariant from the Puiseux expansion with exponents
m = 23 and (a3 , a4 , a5 ) = (33, 48, 78). Since m is prime and coprime to 33, the only
characteristic term is 33 with gcd-sequence (23, 1). The δ-invariant is
1
δq = (33(23 − 1) − 23 + 1) = 352
2
The degree of the secant variety of C is
77
degSecC =
− δp − δq = 2926 − 754 − 352 = 1820
2
4. The degree of the join of two monomial curves
Consider the join of two monomial curves C and D in Pr defined by
C : [s : t] 7→ [sdC : sdC −a1 ta1 : . . . : sdC −ar−1 tar−1 : tdC ]
where a1 < a2 < . . . < ar = dC , and
D : [s : t] 7→ [sdD : sdD −b1 tb1 : . . . : sdD −br−1 tbr−1 : tdD ]
where b1 < b2 < . . . < br = dD . The two sequences
a1 < a2 < . . . < ar = dC ,
b1 < b2 < . . . < br = dD
8
K. RANESTAD
therefore determine the two curves completely. For the parameterizations to be 1 − 1 onto
the image, we ask that the ai have no common factor, and likewise for the bi . The join
is a threefold, so its degree coincides with the number of lines meeting the two curves in
distinct points that also meet a given codimension 3 linear space L in Pr . But this number
equals the number of new intersection points obtained by projecting the union of the two
curves from L to a plane. Denote by πL the projection from L, and let C = πL (C) and
D = πL (D) be the images of C and D respectively. The total intersection number
C · D = dC · dD
by Bezout’s theorem, so to get the degree we have to subtract the intersection multiplicity
at the points of πL (C ∩ D). In our special situation there certainly are points in C ∩ D:
{p = [1 : 0 . . . : 0], q = [0 : · · · : 0 : 1], u = [1 : . . . : 1]} ⊂ C ∩ D.
There may be more:
Lemma 4.1. Let pij = ai bj − aj bi 1 ≤ i < j ≤ r, and let g = gcd{pij )|1 ≤ i < j ≤ r},
then C and D intersect in exactly g points beside p and q, and the intersection in these g
points is transversal. For a general plane projection πL the total intersection multiplicity
of C and D at the points πL (C ∩ D \ {p, q}) is g.
Proof. The intersection points C ∩ D are, besides p and q, precisely the solutions (t1 , t2 )
to the equations
ta1i = tb2i i = 1, . . . , r.
To find these we first consider the equations of the absolute values. Since ti 6= 0, these
are real positive numbers, so we may take logarithms and get a system of linear equations
ai log(|t1 |) = bi log(|t2 |) i = 1, . . . , r.
Now,
gcd(a1 , . . . , ar ) = gcd(b1 , . . . , br ) = 1
and ai is different from bi for some i, so there is at least one nonzero pij . The corresponding
pair of homogeneous equations of logarithms has a unique solution, i.e. only the zerosolution, in particular |t1 | = |t2 | = 1.
Therefore we may write t1 = exp(2πix) = e2πix and t2 = exp(2πiy) = e2πiy , and the
equations translate into the linear conditions
ai x − bi y ∈ Z i = 1, . . . , r
Again, one of the pij must be non-zero, and xpij and ypij are integers. In fact, since
g = gcd{pij )|1 ≤ i < j ≤ r}, the real numbers xg and yg must be integers. Therefore
ai xg − bi yg is an integer divisible by g, so we set X = xg and Y = yg and have reduced
the above equations to the modular equations
Ai X − Bi Y ≡ 0 (g) i = 1, . . . , r,
where Ai ≡ ai (g) and Bi ≡ bi (g). If Ai ≡ 0 (g), then pij ≡ Bi Aj ≡ 0 (g), j 6= i.
Let gi = gcd(bi , g), then gg |aj j = 1, ..., r. But gcd(a1 , . . . , ar ) = 1, so gg = 1 and g|bi .
i
i
By symmetri we get Ai ≡ 0 (g) if only if Bi ≡ 0 (g).
Since gcd(a1 , . . . , ar ) = 1 there is an integral combination of these equations on the
form: X ≡ dY (g), for some d. The above argument applies again to show that d is
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
9
nonzero. Now, any of the pairs (Xα , Yα ) = (−dα, α) α = 1, . . . , g is a solution to this
equation. Since all equations are proportional modulo g these are the solutions to all r
equations. Consequently, the pairs
(exp(
2πiYα
2πiXα
), exp(
)),
g
g
α = 1, ..., g
are the solutions of the original equations. For transversality, we need to check that the
tangent directions of the two curves at intersection points are distinct. At the intersection
points the absolute value of each coordinate is 1, so taking the absolute value of the tangent
directions at these points we get (1, a1 , ..., ar ) and (1, b1 , ..., br ). But these are clearly
distinct by assumption, so transversality follows. Thus at every point of intersection the
two tangents span a plane. A general codimension 3 subspace L ⊂ Pr does not intersect
any of these planes, so the intersections remain transversal after the projection πL to a
plane. At each point of πL (C ∩ D \ {p, q}) the intersection multiplicity is 1, so they add
up to g for the g points.
For the points πL (p) and πL (q) the intersection multiplicity is at least two, since the
two curves have the same tangent(cone) at those points. In fact, since the curves are
unibranched, there is a unique tangent direction at the point, i.e. if they are singular
they have a cusp there. The intersection multiplicity at these points is determined by
a procedure similar to the one given in the previous section. More precisely consider
say the point πL (p). Blow it up and let p1 be the common intersection point of the
strict transforms of the two curves on the exceptional divisor. There is a unique such
intersection point since the two curves are unibranched and the tangents to C and D at
πL (p) coincide. Now blow up in the point p1 . If the strict transforms meet on the new
exceptional divisor, then denote it by p2 and blow up in this point. Continue, until the
strict transforms do not intersect on the exceptional divisor. Thus we get a finite sequence
p0 = πL (p), p1 , . . . , pk , and together with it the multiplicities of the strict transforms of
the two curves at each pi . We denote these multiplicity sequences by m0 (C), . . . , mk (C)
and m0 (D), . . . , mk (D). The intersection multiplicity between the two curves at the point
πL (p) is
IπL (p) (C, D) =
k
X
mi (C)mi (D).
i=0
The multiplicity sequences m0 (C), . . . , mk (C) and m0 (D), . . . , mk (D) are decreasing
and similar to the multiplicity sequences constructed in the previous section. There
are however a main difference in that the we need to consider nonminimal multiplicity
sequences, i.e. sequences that contain 1’s since these terms contribute to the intersection
multiplicity, while they do not contribute to the δ-invariant. Because of the unibranch
property these 1’s will only be appear at the end of the sequences though.
The problem is how to compute these sequences from sequences a1 , ..., ar and b1 , ..., br of
the curves C and D. In this case the non-characteristic terms are as important as the characteristic ones, since the intersection point of the strict transforms with the exceptional
divisor is crucial. Some special cases may illustrate the issue:
10
K. RANESTAD
Example 4.2. Consider monomial curves C : (1, 2, 3, 4) and D : (1, 2, 3, 5). Then the two
curves separate over πL (p) after four blowups and the multiplicity sequences are mi (C) :
1, 1, 1, 1 and mi (D) : 1, 1, 1, 1. The intersection multiplicity at πL (p) is 1 + 1 + 1 + 1 = 4.
Example 4.3. The monomial curves C : (1, 2, 3, 4) and D : (2, 4, 6, 9) separate after four
blowups starting at πL (p), the multiplicity sequences are mi (C) : 1, 1, 1, 1 and mi (D) :
2, 2, 2, 2. The intersection multiplicity at πL (p) is 2 + 2 + 2 + 2 = 8.
Example 4.4. For C : (1, 2, 3, 4) and D : (b0 , b1 , b2 , b3 ) where b0 > 1 and b1 6= 2b0 , then
the strict transforms over πL (p) separate after two blowups and the multiplicities that
contribute to the intersection multiplicity are mi (C) : 1, 1 and mi (D) : b0 , min{(b1 −
b0 ), b0 }. The intersection multiplicity at πL (p) is min{b1 , 2b0 }.
With these examples in mind we formulate the algorithm computing the degree of the
join.
Algorithm 4.5. (Intersection multiplicity algorithm I) Given two monomial curves
C and D defined by the sequences
a1 < a2 < . . . < ar = dC ,
b1 < b2 < . . . < br = dD
respectively, and assume that for some j ≥ 1 bi ≥ ai for i < j while bj > aj . The
following two steps computes the intersection multiplicity of the general projection π of
the two curves to a plane in the point π([1 : 0 : ... : 0]).
Step 1. Let α =
b1
.
a1
If α is not an integer, then set k = 0, otherwise let
k = max{i|bi = αai }
If k ≥ 2, let m1 , m2 , . . . , ms be the multiplicity sequence, the outcome of step 2 of the
algorithm 2.1, of the sequence (a1 , a2 , . . . , ak ), and set
δk = α(m21 + · · · + m2s ).
If k < 2, set δk = 0.
Step 2. If k < 2, apply step 2 of the algorithm 2.1 to the sequences (a1 , a2 ) and (b1 , b2 ),
with outcome
(e1 , r1 ), (e2 , r2 ), ..., (em , rm ) and (e01 , r10 ), ..., (e0n , rn0 )
respectively.
If k ≥ 2, let g = gcd(a1 , ..., ak ), and apply the multiplicity algorithm in section 2.1 to
the sequences (g, ak+1 − ak ) and (gα, bk+1 − bk ), with outcome
(e1 , r1 ), (e2 , r2 ), ..., (em , rm )
and
(e01 , r10 , ..., (e0n , rn0 )
respectively.
Let l = max{j|ei = e0i , i = 1, . . . , j} and let
=
l
X
j
0
0
ej · rj rj0 + e0l+1 rl+1 rl+1
+ rl+1 rl+2
,
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
if
e0l+1
11
< el+1 and
=
l
X
0
0
ej · rj rj0 + el+1 rl+1 rl+1
+ rl+2 rl+1
,
j
if el+1 <
e0l+1 .
Proposition 4.6. The intersection multiplicity between the curves π(C) and π(D) at
π([1 : 0... : 0]) is
I(C, D) = δk + .
Proof. To start we project C and D into the plane and may choose coordinates such that
π(C) and π(D) have the parameterizations
π(C) : x = ta1 + c1,3 ta3 + ... + c1,r tar , y = ta2 + c2,3 ta3 + ... + c2,r tar
and
π(D) : x = tb1 + c1,3 tb3 + ... + c1,r tbr , y = tb2 + c2,3 tb3 + ... + c2,r tbr .
By assumption a1 < a2 and b1 < b2 , so both curves are tangent along the x-axis. Now,
we blow up the plane in the origin. The strict transforms of these curves on the blowup
intersect the exceptional curve in the x-chart (with coordinates (x, xy)). In this chart the
strict transforms π(C)0 and π(D)0 have local parameterizations:
π(C)0 : x = ta1 +c1,3 ta3 +...+c1,r tar , y = ta2 −a1 +c2,3 ta3 −a1 +...+c2,r tar −a1 −c1,3 ta2 +a3 −2a1 +. . .
and
π(D)0 : x = tb1 +c1,3 tb3 +...+c1,r tbr , y = tb2 −b1 +c2,3 tb3 −b1 +...+c2,r tbr −b1 −c1,3 tb2 +b3 −2b1 +. . . .
The tangent at the origin is y = 0 if a1 < a2 − a1 , it is x = 0 if a1 − a2 < a1 and it is
x = y if a1 = a2 − a1 .
Notice, that the terms of order less than ak − a1 and bk − b1 respectively, have the
same coefficients and differ only in the exponent by the factor α. Therefore, if k > 0
the two curves π(C)0 and π(D)0 have the same tangent direction at the origin, and their
strict transform on the blow up in the origin intersect. Proceeding we need to know
after how many blowups, the strict transforms does not intersect, and keep track of the
multiplicities of the two strict transforms up to that point. Computing the number of
blowups needed to separate the two curves, comes down to keeping track of first terms
of the parametrizations of the strict transforms after successive blowups. The tangent
direction decides the parametrization of the strict transform: If the tangent direction is
y = 0 then the strict transform is parametrized by x, xy , if the tangent direction is x = 0,
then the strict transform is parametrized by xy , y, and if the tangent direction is x = y,
then the strict transform is parametrized by x, y−x
. Now, the multiplicities of the strict
x
transforms at the origin form the multiplicity sequence obtained by step 2 of the algorithm
2.1, but keeping track of the tangent directions at each point, we actually also control the
intersection between the two curves. The change from y = 0 to x = 0 of tangent direction
correspond to going from (i, j) to (i, j +1) in the Euclidean algorithm, while the third kind
of tangent corresponds to going from (i, w(i)) to (i+1,1) or to non-characteristic terms.
In this algorithm, as long as i ≤ k, the leading terms of the parametrizations differ only
by a factor of tα . So the corresponding tangent directions coincide. Assume first k ≥ 2. If
12
K. RANESTAD
i = k +1 and j = 1 we have parametrizations tg +..., tak+1 −ak +... and tαg +..., tbk+1 −bk +....
To see when these two curves separate, we apply again the Euclidean algorithm to the
pairs (g, ak+1 − ak ) and (gα, bk+1 − bk ) and get
ak+1 − ak = e1 r1 (= g) + r2 . . . rm = em rm+1
and
0
bk+1 − bk = e01 r10 (= αg) + r20 . . . rn0 = e0n rn+1
So here we compare the coefficients ei and e0i . The curves split after
e1 + e2 + ... + el + min{el+1 , e0l+1 } + 1
blowups if ei = e0i for i ≤ l, while el+1 , 6= e0l+1 }. In fact, the above argument says that
after e1 + e2 + ... + el + min{el+1 , e0l+1 } blowups, the tangent directions of the two strict
transforms still coincides, while after one more blowup they do not, and after two more
blowups the two strict transforms separate.
If k < 2 the number s of blowups needed to separate the two curves is determined by
the initial pairs of exponents (a1 , a2 ) and (b1 , b2 ) and the above procedure starting with
these pairs in stead of (g, ak+1 − ak ) and (gα, bk+1 − bk ) clearly determines s.
The intersection multiplicity algorithm may be shortened.
Lemma 4.7. Let (g, h) and (g 0 , h0 ) be pairs of integers and consider the Euclidean algorithm applied to each pair:
h = e1 r1 + r2 . . . rm = em rm+1
and
0
h0 = e01 r10 + r20 . . . rn0 = e0n rn+1
where r1 = g and r10 = g 0 . Let l = max{j|ei = e0i , i = 1, . . . , j} and set
(P
l
0
0
ej · rj rj0 + e0l+1 rl+1 rl+1
+ rl+1 rl+2
if e0l+1 < el+1
(4.1)
= Pjl
0
0
0
if el+1 < e0l+1 .
j ej · rj rj + el+1 rl+1 rl+1 + rl+2 rl+1
Then
(4.2)
(
gh0 if l is even and e0l+1 < el+1 , or l is odd and e0l+1 > el+1
=
g 0 h if l is odd and e0l+1 < el+1 , or l is even and e0l+1 > el+1 .
P
0
Proof. Rewrite the sum lj ej · rj rj0 using the identity (ri−1 − ri+1 )ri0 = ei ri ri0 or (ri−1
−
0
0 0
ri+1 )ri = ei ri ri from the Euclidean algorithm.
Algorithm 4.8. (Intersection multiplicity II) Given two monomial curves C and D
defined by the sequences
A : a1 < a2 < . . . < ar = dC ,
respectively. Let α =
b1
.
a1
B : b1 < b2 < . . . < br = dD
If α is not an integer, then set k = 0, otherwise let
k = max{i|bi = αai }
THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES
13
If k < 2, consider the Euclidean algorithm applied to the pairs (a1 , a2 ) and (b1 , b2 ), with
factors and residues
(e1 , r1 (= a1 )), (e2 , r2 ), ..., (em , rm )
(e01 , r10 (= b1 )), ..., (e0n , rn0 )
and
respectively. Let l = max{j|ei = e0i , i = 1, . . . , j}.
Set
(
a1 b2
=
a2 b1
(4.3)
if l is even and e0l+1 < el+1 or l is odd and el+1 < e0l+1
if l is even and e0l+1 > el+1 or l is odd and el+1 > e0l+1 .
If k ≥ 2, let
g1 = a1 ,
gi = gcd{a1 , ..., ai }
i = 2, . . . , k
and consider the Euclidean algorithm applied to the pairs (gk , ak+1 − ak ) and (αgk , bk+1 −
bk ), with factors and residues
(e1 , r1 (= gk )), (e2 , r2 ), ..., (em , rm )
and
(e01 , r10 (= αgk )), ..., (e0n , rn0 )
respectively. Let l = max{j|ei = e0i , i = 1, . . . , j}.
Set
(
gk (bk+1 − bk )
if l is even and e0l+1 < el+1 or l is odd and e0l+1 > el+1
(4.4) =
αgk (ak+1 − ak ) if l is even and e0l+1 > el+1 or l is odd and el+1 > e0l+1 .
Then the intersection multiplicity of the general plane projection of the two curves C
and D, given by the sequences A and B respectively, at the image of p = [1 : 0... : 0] is
(
if k < 2
Pk−1
(4.5)
I(A, B) =
α( i=2 ai (gi−1 − gi ) + ak gk−1 ) + if k ≥ 2.
Proof. The proof follows the argument of [2, p. 194, ex. 5.6] applied to the algorithm 4.5.
According to Proposition 4.6 the intersection multiplicity at π([1 : 0... : 0]) is
I(C, D) = α(m21 + · · · + m2s ) + ,
where m1 , . . . , ms is the multiplicity sequence of the sequence {a1 < ... < ak }. The sum
s
X
i=1
αm2i
=α
s
X
m2i
i=1
may now be computed as in the proof of Proposition 2.2, i.e.
α
s
X
i=1
m2i
= αa2 g1 + α
k−1
X
i=2
k−1
X
ai+1 − ai )gi = α( (ai+1 (gi − gi+1 ) + ak gk−1 ).
i=1
For the residual contribution we apply the Lemma 4.7. So the intersection multiplicity
I(C, D) coincides with I(A, B).
Using 4.8 to compute I(A, B) for a pair of sequences A and B we conclude:
14
K. RANESTAD
Proposition 4.9. Given two monomial curves C and D defined by the sequences
A : a1 < a2 < . . . < ar = dC ,
B : b1 < b2 < . . . < br = dD
respectively. Set
A0 : ar −ar−1 < ar −ar−2 < . . . < ar −a1 < dC ,
B 0 : br −br−1 < br −br−2 < . . . < br −b1 < dD ,
and let g = gcd{ai bj − aj bi |1 ≤ i < j ≤ r}. Then the degree of the join of C and D is
degJoin(C, D) = dC · dD − I(A, B) − I(A0 , B 0 ) − g.
References
[1] E. Brieskorn and H. Knörrer: Plane algebraic curves. Birkhäuser, 1986
[2] E. Casas-Alvero: “Plane curve singularities.” London Math. Soc. Lecture Notes 276, Cambridge
University Press, 2000
[3] R. Hartshorne: Algebraic Geometry. GTM, Springer Verlag, New York, 1977
Matematisk Institutt, Universitetet i Oslo P.O.Box 1053, Blindern, N-0316 Oslo,
NORWAY
E-mail address: ranestad@math.uio.no